Cycle Representations of Markov Processes

2. Auflage 2007
ISBN: 978-0-387-36081-2
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Wasserzeichen (»Systemvoraussetzungen)

E-Book, Englisch, 304 Seiten

"This book is a prototype providing new insight into Markovian dependence via the cycle decompositions. It presents a systematic account of a class of stochastic processes known as cycle (or circuit) processes - so-called because they may be defined by directed cycles. These processes have special and important properties through the interaction between the geometric properties of the trajectories and the algebraic characterization of the Markov process. An important application of this approach is the insight it provides to electrical networks and the duality principle of networks. In particular, it provides an entirely new approach to infinite electrical networks and their applications in topics as diverse as random walks, the classification of Riemann surfaces, and to operator theory. The second edition of this book adds new advances to many directions, which reveal wide-ranging interpretations of the cycle representations like homologic decompositions, orthogonality equations, Fourier series, semigroup equations, and disintegration of measures. The versatility of these interpretations is consequently motivated by the existence of algebraic-topological principles in the fundamentals of the cycle representations. This book contains chapter summaries as well as a number of detailed illustrations. Review of the earlier edition: ""This is a very useful monograph which avoids ready ways and opens new research perspectives. It will certainly stimulate further work, especially on the interplay of algebraic and geometrical aspects of Markovian dependence and its generalizations."" Math Reviews."

Autoren/Hrsg.

Inhaltsverzeichnis

1;Preface to the Second Edition;7
2;Preface;9
3;Acknowledgments;15
4;Contents;16
5;Fundamentals of the Cycle Representations of Markov Processes;20
5.1;Directed Circuits;21
5.1.1;1.1 Definition of Directed Circuits;22
5.1.2;1.2 The Passage Functions;26
5.1.3;1.3 Cycle Generating Equations;28
5.2;Genesis of Markov Chains by Circuits: The Circuit Chains;35
5.2.1;2.1 Finite Markov Chains Defined by Weighted Circuits;35
5.2.2;2.2 Denumerable Markov Chains Generated by Circuits;41
5.3;Cycle Representations of Recurrent Denumerable Markov Chains;47
5.3.1;3.1 The Derived Chain of Qians;47
5.3.2;3.2 The Circulation Distribution of a Markov Chain;53
5.3.3;3.3 A Probabilistic Cycle Decomposition for Recurrent Markov Chains;55
5.3.4;3.4 Weak Convergence of Sequences of Circuit Chains: A Deterministic Approach;57
5.3.5;3.5 Weak Convergence of Sequences of Circuit Chains: A Probabilistic Approach;63
5.3.6;3.6 The Induced Circuit Chain;65
5.4;Circuit Representations of Finite Recurrent Markov Chains;72
5.4.1;4.1 Circuit Representations by Probabilistic Algorithms;73
5.4.2;4.2 Circuit Representations by Nonrandomized Algorithms;74
5.4.3;4.3 The Caratheodory-Type Circuit Representations;77
5.4.4;4.4 The Betti Number of a Markov Chain;78
5.4.5;4.5 A Refined Cycle Decomposition of Finite Stochastic Matrices: A Homologic Approach;83
5.4.6;4.6 The Dimensions of Caratheodory and Betti;89
5.5;Continuous Parameter Circuit Processes with Finite State Space;90
5.5.1;5.1 Genesis of Markov Processes by Weighted Circuits;90
5.5.2;5.2 The Weight Functions;93
5.5.3;5.3 Continuity Properties of the Weight Functions;96
5.5.4;5.4 Differentiability Properties of the Weight Functions;100
5.5.5;5.5 Cycle Representation Theorem for Transition Matrix Functions;102
5.5.6;5.6 Cycle Representation Theorem for Q-Matrices;105
5.6;Spectral Theory of Circuit Processes;109
5.6.1;6.1 Unitary Dilations in Terms of Circuits;109
5.6.2;6.2 Integral Representations of the Circuit-Weights Decomposing Stochastic Matrices;112
5.6.3;6.3 Spectral Representation of Continuous Parameter Circuit Processes;114
5.7;Higher-Order Circuit Processes;117
5.7.1;7.1 Higher-Order Markov Chains;117
5.7.2;7.2 Higher-Order Finite Markov Chains Defined by Weighted Circuits;122
5.7.3;7.3 The Rolling-Circuits;133
5.7.4;7.4 The Passage-Function Associated with a Rolling- Circuit;136
5.7.5;7.5 Representation of Finite Multiple Markov Chains by Weighted Circuits;139
5.8;Cycloid Markov Processes;147
5.8.1;8.1 The Passages Through a Cycloid;147
5.8.2;8.2 The Cycloid Decomposition of Balanced Functions;151
5.8.3;8.3 The Cycloid Transition Equations;153
5.8.4;8.4 Definition of Markov Chains by Cycloids;157
5.9;Markov Processes on Banach Spaces on Cycles;160
5.9.1;9.1 Banach Spaces on Cycles;160
5.9.2;9.2 Fourier Series on Directed Cycles;167
5.9.3;9.3 Orthogonal Cycle Transforms for Finite Stochastic Matrices;172
5.9.4;9.4 Denumerable Markov Chains on Banach Spaces on Cycles;176
5.10;The Cycle Measures;178
5.10.1;10.1 The Passage-Functions as Characteristic Functions;178
5.10.2;10.2 The Passage-Functions as Balanced Functions;182
5.10.3;10.3 The Vector Space Generated by the Passage- Functions;186
5.10.4;10.4 The Cycle Measures;189
5.10.5;10.5 Measures on the Product of Two Measurable Spaces by Cycle Representations of Balanced Functions: A Fubini- Type Theorem;196
5.11;Wide-Ranging Interpretations of the Cycle Representations of Markov Processes;201
5.11.1;11.1 The Homologic Interpretation of the Cycle Processes;201
5.11.2;11.2 An Algebraic Interpretation;206
5.11.3;11.3 The Banach Space Approach;208
5.11.4;11.4 The Measure Theoretic Interpretation;209
5.11.5;11.5 The Cycle Representation Formula as a Disintegration of Measures;211
6;Applications of the Cycle Representations;218
6.1;Stochastic Properties in Terms of Circuits;219
6.1.1;1.1 Recurrence Criterion in Terms of the Circuits;219
6.1.2;1.2 The Entropy Production of Markov Chains;222
6.1.3;1.3 Reversibility Criteria in Terms of the Circuits;224
6.1.4;1.4 Derriennic Recurrence Criterions in Terms of the Weighted Circuits;227
6.2;Levy’s Theorem Concerning Positiveness of Transition Probabilities;236
6.2.1;2.1 Levy’s Theorem in Terms of Circuits;237
6.2.2;2.2 Physical Interpretation of the Weighted Circuits Representing a Markov Process;239
6.3;The Rotational Theory of Markov Processes;242
6.3.1;3.1 Preliminaries;242
6.3.2;3.2 Joel E. Cohen’s Conjecture on Rotational Representations of Stochastic Matrices;245
6.3.3;3.3 Alpern’s Solution to the Rotational Problem;246
6.3.4;3.4 Transforming Circuits into Circle Arcs;251
6.3.5;3.5 Mapping Stochastic Matrices into Partitions and a Probabilistic Solution to the Rotational Problem;258
6.3.6;3.6 The Rotational Dimension of Stochastic Matrices and a Homologic Solution to the Rotational Problem;261
6.3.7;3.7 The Complexity of the Rotational Representations;266
6.3.8;3.8 A Reversibility Criterion in Terms of Rotational Representations;270
6.3.9;3.9 Rotational Representations of Transition Matrix Functions;273
7;List of Notations;276
8;Bibliography;278
9;Index;308